On the Complexity of the Cogrowth Sequence

Given a finitely generated group with generating set $S$, we study the \emph{cogrowth} sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks in a Cayley graph with steps from $S$. We prove that the cogrowth sequence is not $P$-recursive when~$G$ is an amenable group of superpolynomial growth, answering a question of Garrabant and Pak.